Non-linear incidence and stability of infectious disease models

Korobeinikov, Andrei; Maini, Philip K.

doi:10.1093/imammb/dqi001

Cited by 258 publications

(200 citation statements)

References 21 publications

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“…Li et al (1999) studied the global properties of a SEIR model with the incidence rate of the form g(I)S; they also extended some of their results for a SEIRS model. Korobeinikov and Maini (2005) considered a variety of models with the incidence rate of the form g(I)h(S); constructed Lyapunov functions enabled to establish global properties for a SIR and a SEIR models. The most general case was considered by Feng and Thieme (2000,a), who considered properties of a model with arbitrarily many stages of infection, all of which have general length distributions, and with incidence rate of a very general form.…”

Section: Introductionmentioning

confidence: 99%

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Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

2006

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Section: Introductionmentioning

confidence: 99%

“…In this note we attempt to extend the Lyapunov functions constructed earlier (Korobeinikov and Maini, 2005) to a more general incidence rate given by an arbitrary function f (S, I). We prove that the concavity of the incidence rate with respect to the number of the infective hosts leads to a globally stable system.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

2006

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“…We shall construct two suitable Lyapunov functionals which are modified from those in [3,4,6,7,9]. Our result is more decent in the sense that no additional condition is required to obtain global stability of endemic equilibrium; while in the literature (cf.…”

Section: P(τ ) E −μτ βS(t − τ )I(t − τ ) Dτ − μ I I(t) − γ I(t)mentioning

confidence: 99%

Global stability analysis of a delayed susceptible–infected–susceptible epidemic model

Paulhus¹,

Wang²

2014

Journal of Biological Dynamics

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We study a susceptible-infected-susceptible model with distributed delays. By constructing suitable Lyapunov functionals, we demonstrate that the global dynamics of this model is fully determined by the basic reproductive ratio R 0 . To be specific, we prove that if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable. On the other hand, if R 0 > 1, then the endemic equilibrium is globally asymptotically stable. It is remarkable that the model dynamics is independent of the probability of immunity lost.

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“…Both of these models were comprehensively investigated recently [48][49][50][51]53], and their properties were established. It was rigorously proved that the condition ∂ 2 f ∂I 2 ≤ 0 is sufficient to ensure the global stability 209…”

Section: Memory Componentmentioning

confidence: 99%

Memory Effects in Population Dynamics : Spread of Infectious Disease as a Case Study

Pimenov

Kelly

Korobeinikov³

et al. 2012

Math. Model. Nat. Phenom.

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Abstract. Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects.

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Non-linear incidence and stability of infectious disease models

Cited by 258 publications

References 21 publications

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Global stability analysis of a delayed susceptible–infected–susceptible epidemic model

Memory Effects in Population Dynamics : Spread of Infectious Disease as a Case Study

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